Given
[tex]f(x)=\frac{x+6}{x-7}[/tex]
Recall
The horizontal line test can be used to determine if a function is one-to-one given a graph. Simply superimpose a horizontal line onto a graph and see if it intersects the graph at more than one point. If it does, the graph is not one-to-one and if it only intersects at one point, it will be one-to-one.
The graph
It passed the horizontal line test, therefore is one to one function
Part B
[tex]f(x)=\frac{x+6}{x-7}[/tex]
Step 1
Replace f(x) with y
[tex]y=\frac{x+6}{x-7}[/tex]
Step 2
Inter change y and x
[tex]x=\frac{y+6}{y-7}[/tex]
Step 3
Make y the subject
[tex]\begin{gathered} x=\frac{y+6}{y-7} \\ x(y-7)=y+6 \\ xy-7x=y+6 \\ xy-y=6+7x \\ y(x-1)=6+7x \\ divide\text{ both sides by x-1} \\ y=\frac{6+7x}{x-1} \end{gathered}[/tex]
Step 4
Replace y with f^-1
[tex]f^{-1}(x)=\frac{6+7x}{x-1}[/tex]
The final answer
[tex]f^{-1}(x)=\frac{6+7x}{x-1}[/tex]
